To prove the given trigonometric equation, we can start by manipulating the left-hand side of the equation using trigonometric identities.

Recall the Pythagorean identity:
cos^2(x) + sin^2(x) = 1

Now, let's begin by squaring the given equation:
(cos^2x - sin^2x)^2 = (2cosx - 1)^2
Expanding the left-hand side:
cos^4x - 2cos^2xsin^2x + sin^4x = 4cos^2x - 4cosx + 1
Since cos^2x + sin^2x = 1, we can rewrite cos^4x - 2cos^2xsin^2x + sin^4x as:
1 - 2cos^2xsin^2x
Substitute this back into the equation:
1 - 2cos^2xsin^2x = 4cos^2x - 4cosx + 1
Now, simplify the equation:

2cos^2xsin^2x = 4cos^2x - 4cosx
Rearrange the equation and factor out a 2cosx:
2cosx(2cosx - 1) = -2cos^2xsin^2x
Now, we can use the Pythagorean identity to simplify the right-hand side of the equation:
2cosx(2cosx - 1) = -2cos^2x(1 - cos^2x)
2cosx(2cosx - 1) = -2cos^2xsin^2x
Therefore, we have proven that cos^2x - sin^2x = 2cosx - 1.